Inverse Functions

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Answer Key & Explanations

Inverse Functions · Grade 12 · Worksheet 1

1. A biologist is modeling the population growth of a rare species using the function P(t) = 500e^(0.03t), where t represents time in years and P(t) represents the population size. To determine how long it will take for the population to reach a specific target size, the biologist needs to find the inverse function. Find the inverse function t(P) that gives the time required to reach population P. Answer: t(P) = (ln(P) - ln(500))/0.03 Solution: P(t) = 500 * e^(0.03 * t) We want the inverse function t(P), which tells us the time needed to reach population P.
   Full step-by-step solution

   We start with the population function: P(t) = 500 * e^(0.03 * t) We want the inverse function t(P), which tells us the time needed to reach population P. Step 1: Write the equation with P instead of P(t): P = 500 * e^(0.03 * t) Step 2: Isolate the exponential term by dividing both sides by 500: P / 500 = e^(0.03 * t) Step 3: To solve for t, we take the natural logarithm (ln) of both sides. This is because ln and e are inverse functions. ln(P / 500) = ln(e^(0.03 * t)) Step 4: Use the logarithm property: ln(e^x) = x. So the right-hand side becomes: ln(P / 500) = 0.03 * t Step 5: Solve for t by dividing both sides by 0.03: t = ln(P / 500) / 0.03 Step 6: Use the logarithm property: ln(a / b) = ln(a) - ln(b). So we can write: t = [ln(P) - ln(500)] / 0.03 This is the inverse function t(P). Final answer: t(P) = (ln(P) - ln(500)) / 0.03

2. A biomedical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (2t + 3)/(t - 1), where C represents concentration in mg/L and t represents time in hours. To determine when the concentration reaches a specific level, they need to find the inverse function. What is the inverse function t(C) that gives the time when the concentration is C mg/L? Answer: t(C) = (C + 3)/(C - 2) Solution: C(t) = (2t + 3)/(t - 1) We want to find the inverse function t(C), so we need to solve for t in terms of C. Replace C(t) with C. C = (2t + 3)/(t - 1) Multiply both sides by (t - 1) to eliminate the denominator.
   Full step-by-step solution

   We start with the concentration function: C(t) = (2t + 3)/(t - 1) We want to find the inverse function t(C), so we need to solve for t in terms of C. Step 1: Replace C(t) with C. C = (2t + 3)/(t - 1) Step 2: Multiply both sides by (t - 1) to eliminate the denominator. C(t - 1) = 2t + 3 Step 3: Distribute C on the left side. Ct - C = 2t + 3 Step 4: Bring all terms with t to one side and constant terms to the other side. Ct - 2t = C + 3 Step 5: Factor t from the left side. t(C - 2) = C + 3 Step 6: Solve for t by dividing both sides by (C - 2). t = (C + 3)/(C - 2) Step 7: Write the inverse function notation. t(C) = (C + 3)/(C - 2) This is the inverse function that gives the time t when the concentration is C mg/L.

3. A function is represented graphically as a cubic curve with inflection point at (1, 2) and passing through points (0, 3) and (2, 1). The function has the form f(x) = a(x - h)³ + k. Find the algebraic expression for its inverse function f⁻¹(x). Answer: f⁻¹(x) = 1 + ∛(x - 2) Solution: Identify the vertex form: f(x) = a(x - 1)³ + 2 Use point (0, 3) to find 'a': 3 = a(0 - 1)³ + 2 3 = a(-1)³ + 2 3 = -a + 2 -a = 1, so a = -1 The function is f(x) = -(x - 1)³ + 2 To find the inverse, swap x and y: x = -(y - 1)³ + 2 Solve for y: x - 2 = -(y - 1)³ Multiply both sides by -1: 2 - x =…
   Full step-by-step solution

   Step 1: Identify the vertex form: f(x) = a(x - 1)³ + 2 Step 2: Use point (0, 3) to find 'a': 3 = a(0 - 1)³ + 2 Step 3: 3 = a(-1)³ + 2 Step 4: 3 = -a + 2 Step 5: -a = 1, so a = -1 Step 6: The function is f(x) = -(x - 1)³ + 2 Step 7: To find the inverse, swap x and y: x = -(y - 1)³ + 2 Step 8: Solve for y: x - 2 = -(y - 1)³ Step 9: Multiply both sides by -1: 2 - x = (y - 1)³ Step 10: Take cube root: ∛(2 - x) = y - 1 Step 11: Add 1 to both sides: y = 1 + ∛(2 - x) Step 12: Write in standard form: f⁻¹(x) = 1 + ∛(2 - x) The inverse function is f⁻¹(x) = 1 + ∛(2 - x)

4. An environmental scientist is modeling the decay of a radioactive isotope used in carbon dating. The remaining mass M(t) in grams after t years is given by M(t) = 100 * (1/2)^(t/5730). To determine how long it takes for the isotope to decay to a specific mass, the scientist needs to find the inverse function. What is the inverse function t(M) that gives the time required for the mass to reach M grams? Answer: t(M) = 5730 * log2(100/M) Solution: Start with the original function: M = 100 * (1/2)^(t/5730) Divide both sides by 100: M/100 = (1/2)^(t/5730) Take the logarithm base 2 of both sides: log2(M/100) = log2((1/2)^(t/5730)) Use the power rule for logarithms: log2(M/100) = (t/5730) * log2(1/2) Since log2(1/2) = -1, we get: log2(M/100)…
   Full step-by-step solution

   Step 1: Start with the original function: M = 100 * (1/2)^(t/5730) Step 2: Divide both sides by 100: M/100 = (1/2)^(t/5730) Step 3: Take the logarithm base 2 of both sides: log2(M/100) = log2((1/2)^(t/5730)) Step 4: Use the power rule for logarithms: log2(M/100) = (t/5730) * log2(1/2) Step 5: Since log2(1/2) = -1, we get: log2(M/100) = -t/5730 Step 6: Multiply both sides by -1: -log2(M/100) = t/5730 Step 7: Use the property -log2(a) = log2(1/a): log2(100/M) = t/5730 Step 8: Multiply both sides by 5730: t = 5730 * log2(100/M) The inverse function is t(M) = 5730 * log2(100/M)

5. f(x) = (2x - 3)/(x + 1), find f⁻¹(x) = ? Answer: f⁻¹(x) = (x + 3)/(2 - x) Solution: f(x) = (2x - 3)/(x + 1) y = (2x - 3)/(x + 1) x = (2y - 3)/(y + 1) Multiply both sides by (y + 1): x(y + 1) = 2y - 3 xy + x = 2y - 3 xy - 2y = -3 - x y(x - 2) = -3 - x y = (-3 - x)/(x - 2) Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) f⁻¹(x) = (x + 3)/(2 - x) f⁻¹(x) = (x + 3)/(2 - x)
   Full step-by-step solution

   Let's find the inverse function step by step. We are given: f(x) = (2x - 3)/(x + 1) --- **Step 1: Replace f(x) with y** y = (2x - 3)/(x + 1) --- **Step 2: Swap x and y** To find the inverse, we swap x and y: x = (2y - 3)/(y + 1) --- **Step 3: Solve for y** Multiply both sides by (y + 1): x(y + 1) = 2y - 3 Expand left side: xy + x = 2y - 3 --- **Step 4: Get all terms with y on one side** xy - 2y = -3 - x Factor y on the left: y(x - 2) = -3 - x --- **Step 5: Solve for y** y = (-3 - x)/(x - 2) --- **Step 6: Simplify** Multiply numerator and denominator by -1: y = (x + 3)/(2 - x) --- **Step 7: Write final inverse function** f⁻¹(x) = (x + 3)/(2 - x) --- **Final answer:** f⁻¹(x) = (x + 3)/(2 - x)

6. Matiu is a marine biologist modeling the water temperature in a coastal research station. The temperature T(t) in degrees Celsius at time t hours after sunrise is given by the rational function T(t) = (6t + 8)/(t - 2). To predict when the temperature will reach a specific value, Matiu needs to find the inverse function that expresses time as a function of temperature. Determine the inverse function t(T). Answer: t(T) = (2T + 8)/(T - 6) Solution: Start with the original function T(t) = (6t + 8)/(t - 2). Replace T(t) with y: y = (6t + 8)/(t - 2). Swap the variables (t becomes y and y becomes T): t = (6y + 8)/(y - 2).
   Full step-by-step solution

   Step 1: Start with the original function T(t) = (6t + 8)/(t - 2). Step 2: Replace T(t) with y: y = (6t + 8)/(t - 2). Step 3: Swap the variables (t becomes y and y becomes T): t = (6y + 8)/(y - 2). Step 4: Multiply both sides by (y - 2): t(y - 2) = 6y + 8. Step 5: Distribute t on the left: ty - 2t = 6y + 8. Step 6: Bring terms with y to one side and constants to the other: ty - 6y = 2t + 8. Step 7: Factor out y on the left: y(t - 6) = 2t + 8. Step 8: Solve for y: y = (2t + 8)/(t - 6). Step 9: Replace y with t(T) and t with T: t(T) = (2T + 8)/(T - 6). The inverse function is t(T) = (2T + 8)/(T - 6).